Shannon–McMillan–Breiman Theorem Along Almost Geodesics in Negatively Curved Groups

Abstract

Consider a non-elementary Gromov-hyperbolic group $\Gamma$ with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on $(X,\mu )$. We construct special increasing sequences of finite subsets $F_n(y)\subset \Gamma$, with $(Y,\nu )$ a suitable probability space, with the following properties.

• Given any countable partition $\mathcal {P}$ of X of finite Shannon entropy, the refined partitions $\bigvee _{\gamma \in F_n(y)}\gamma \mathcal {P}$ have normalized information functions which converge to a constant limit, for $\mu$-almost every $x\in X$ and $\nu$-almost every $y\in Y$.

• The sets $\mathcal {F}_n(y)$ constitute almost-geodesic segments, and $\bigcup _{n\in \mathbb {N}} F_n(y)$ is a one-sided almost geodesic with limit point $F^+(y)\in \partial \Gamma$, starting at a fixed bounded distance from the identity, for almost every $y\in Y$.

• The distribution of the limit point $F^+(y)$ belongs to the Patterson–Sullivan measure class on $\partial \Gamma$ associated with the invariant hyperbolic metric.

The main result of the present paper amounts therefore to a Shannon–McMillan–Breiman theorem along almost-geodesic segments in any p.m.p. action of $\Gamma$ as above. For several important classes of examples we analyze, the construction of $F_n(y)$ is purely geometric and explicit. Furthermore, consider the infimum of the limits of the normalized information functions, taken over all $\Gamma$-generating partitions of X. Using an important inequality due to Seward (Weak containment and Rokhlin entropy, arxiv:1602.06680, 2016), we deduce that it is equal to the Rokhlin entropy $\mathfrak {h}^{\text {Rok}}$ of the $\Gamma$-action on $(X,\mu )$ defined in Seward (Invent Math 215:265–310, 2019), provided that the action is free. Remarkably, this property holds for every choice of invariant hyperbolic metric, every choice of suitable auxiliary space $(Y,\nu )$ and every choice of special family $F_n(y)$ as above. In particular, for every $\epsilon > 0$, there is a generating partition $\mathcal {P}_\epsilon$, such that for almost every $y\in Y$, the partition refined using the sets $F_n(y)$ has most of its atoms of roughly constant measure, comparable to $\exp (-n\mathfrak {h}^{\text {Rok}}\pm \epsilon )$. This describes an approximation to the Rokhlin entropy in geometric and dynamical terms, for actions of word-hyperbolic groups.